Fractional Kirchhoff type equations involving Neumann conditions

In this paper, we are interested in approaching a new class of fractional $ s(\xi) $-Kirchhoff equations with Neumann boundary conditions. First, we discuss properties of the fractional $ s(\xi) $-Kirchhoff equation via the Fatou lemma and the Minty-Browder theorem. In this sense, we attack the existence and multiplicity of solutions to the main problem through a variational approach and the theory of the variable exponent $ \psi $ fractional space, under appropriate assumptions on $ f $ and $ \mathcal{M} $ involved in the $ s(\xi) $-Kirchhoff equation. Finally, we discuss the result with concave and convex nonlinearities.